naginterfaces.library.mv.discrim

naginterfaces.library.mv.discrim(x, isx, ing, ng, wt=None)

discrim computes a test statistic for the equality of within-group covariance matrices and also computes matrices for use in discriminant analysis.

For full information please refer to the NAG Library document for g03da

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g03/g03daf.html

Parameters

xfloat, array-like, shape $(n,m)$

$x[\text{k}-1,\text{l}-1]$ must contain the $\text{k}$th observation for the $\text{l}$th variable, for $\text{l}=1,2,\dots ,m$, for $\text{k}=1,2,\dots ,n$.

isxint, array-like, shape $\left(m\right)$

$isx[l-1]$ indicates whether or not the $l$th variable in $x$ is to be included in the variance-covariance matrices.

If $isx[\text{l}-1]>0$ the $\text{l}$th variable is included, for $\text{l}=1,2,\dots ,m$; otherwise it is not referenced.

ingint, array-like, shape $\left(n\right)$

$ing[\text{k}-1]$ indicates to which group the $\text{k}$th observation belongs, for $\text{k}=1,2,\dots ,n$.

ngint

The number of groups, $n_g$.

wtNone or float, array-like, shape $\left(n\right)$, optional

The elements of $wt$ must contain the weights to be used in the analysis and the effective number of observations for a group is the sum of the weights of the observations in that group. If $wt[k-1]=0.0$ then the $k$th observation is excluded from the calculations.

If weights are not provided then $wt$ must be set to None and the effective number of observations for a group is the number of observations in that group.

Returns

nigint, ndarray, shape $\left(ng\right)$

$nig[\text{j}-1]$ contains the number of observations in the $\text{j}$th group, for $\text{j}=1,2,\dots ,n_g$.

gmnfloat, ndarray, shape $(ng,\text{nvar})$

The $\text{j}$th row of $gmn$ contains the means of the $p$ selected variables for the $\text{j}$th group, for $\text{j}=1,2,\dots ,n_g$.

detfloat, ndarray, shape $\left(ng\right)$

The logarithm of the determinants of the within-group variance-covariance matrices.

gcfloat, ndarray, shape $((ng+1)\times \text{nvar}\times (\text{nvar}+1)/2)$

The first $p(p+1)/2$ elements of $gc$ contain $R$ and the remaining $n_g$ blocks of $p(p+1)/2$ elements contain the $R_j$ matrices. All are stored in packed form by columns.

statfloat

The likelihood-ratio test statistic, $G$.

dffloat

The degrees of freedom for the distribution of $G$.

sigfloat

The significance level for $G$.

Raises

NagValueError

(errno $1$)

On entry, $\text{weight}=⟨value⟩$.

Constraint: $\text{weight}=\text{'U'}$ or $\text{'W'}$.

(errno $1$)

On entry, $m=⟨value⟩$ and $\text{nvar}=⟨value⟩$.

Constraint: $m\ge \text{nvar}$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $1$)

On entry, $ng=⟨value⟩$.

Constraint: $ng\ge 2$.

(errno $1$)

On entry, $\text{nvar}=⟨value⟩$.

Constraint: $\text{nvar}\ge 1$.

(errno $2$)

On entry, $i=⟨value⟩$ and $wt[i-1]<0.0$.

Constraint: $wt[i-1]\ge 0.0$.

(errno $3$)

The number of observations for group $⟨value⟩$ is less than $\text{nvar}$.

(errno $3$)

The effective number of observations for group $⟨value⟩$ is less than $1$.

(errno $3$)

On entry, $i=⟨value⟩$, $ing[i-1]=⟨value⟩$ and $ng=⟨value⟩$.

Constraint: $1\le ing[i-1]\le ng$.

(errno $3$)

On entry, $\text{nvar}=⟨value⟩$ and $⟨value⟩$ values of $isx>0$

Constraint: exactly $\text{nvar}$ elements of $isx>0$.

(errno $4$)

$R$ is not of full rank.

(errno $4$)

$R_j$ is not of full rank for $j=⟨value⟩$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

Let a sample of $n$ observations on $p$ variables come from $n_g$ groups with $n_j$ observations in the $j$th group and $\sum n_j=n$. If the data is assumed to follow a multivariate Normal distribution with the variance-covariance matrix of the $j$th group ${\Sigma }_j$, then to test for equality of the variance-covariance matrices between groups, that is, ${\Sigma }_1={\Sigma }_2=\cdots ={\Sigma }_{n_g}=\Sigma$, the following likelihood-ratio test statistic, $G$, can be used;

$$G=C\{(n-n_g)\log \left(\left|S\right|\right)-\sum _{j=1}^{n_g}(n_j-1)\log \left(\left|S_j\right|\right)\}\text{,}$$

where

$$C=1-\frac{2p^2+3p-1}{6(p+1)(n_g-1)}(\sum _{j=1}^{n_g}\frac{1}{(n_j-1)}-\frac{1}{(n-n_g)})\text{,}$$

and $S_j$ are the within-group variance-covariance matrices and $S$ is the pooled variance-covariance matrix given by

$$S=\frac{{\sum }_{j=1}^{n_g}(n_j-1)S_j}{(n-n_g)}\text{.}$$

For large $n$, $G$ is approximately distributed as a ${\chi }^2$ variable with $\frac{1}{2}p(p+1)(n_g-1)$ degrees of freedom, see Morrison (1967) for further comments. If weights are used, then $S$ and $S_j$ are the weighted pooled and within-group variance-covariance matrices and $n$ is the effective number of observations, that is, the sum of the weights.

Instead of calculating the within-group variance-covariance matrices and then computing their determinants in order to calculate the test statistic, discrim uses a $QR$ decomposition. The group means are subtracted from the data and then for each group, a $QR$ decomposition is computed to give an upper triangular matrix $R_j^{\ast }$. This matrix can be scaled to give a matrix $R_j$ such that $S_j=R_j^T{R}_j$. The pooled $R$ matrix is then computed from the $R_j$ matrices. The values of $\left|S\right|$ and the $\left|S_j\right|$ can then be calculated from the diagonal elements of $R$ and the $R_j$.

This approach means that the Mahalanobis squared distances for a vector observation $x$ can be computed as $z^{T}z$, where $R_{j}z=(x-{\overline{x}}_j)$, ${\overline{x}}_j$ being the vector of means of the $j$th group. These distances can be calculated by discrim_mahal(). The distances are used in discriminant analysis and discrim_group() uses the results of discrim to perform several different types of discriminant analysis. The differences between the discriminant methods are, in part, due to whether or not the within-group variance-covariance matrices are equal.

References

Aitchison, J and Dunsmore, I R, 1975, Statistical Prediction Analysis, Cambridge

Kendall, M G and Stuart, A, 1976, The Advanced Theory of Statistics (Volume 3), (3rd Edition), Griffin

Krzanowski, W J, 1990, Principles of Multivariate Analysis, Oxford University Press

Morrison, D F, 1967, Multivariate Statistical Methods, McGraw–Hill

See also

naginterfaces.library.examples.mv.discrim_group_ex.main()